Problem: Simplify the following expression and state the condition under which the simplification is valid: $t = \dfrac{z^2 + 8z + 7}{z^2 + 7z}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{z^2 + 8z + 7}{z^2 + 7z} = \dfrac{(z + 1)(z + 7)}{(z)(z + 7)} $ Notice that the term $(z + 7)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(z + 7)$ gives: $t = \dfrac{z + 1}{z}$ Since we divided by $(z + 7)$, $z \neq -7$. $t = \dfrac{z + 1}{z}; \space z \neq -7$